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#72267835

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC22×#384 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#11715690
commutator, cosocleC2×C22⋊(C42⋊C3)C22
FrattiniC23C2×#1020
FittingC22×#384D3
socleC25C2×S4
hypocenterC22⋊(C42⋊C3)D4
LCSG ▷ C2×C22⋊(C42⋊C3) ▷ C22⋊(C42⋊C3)
UCSC1 ◁ C2

1527 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 1527 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 20 points:

(1,4,3)(5,6,7,8)(9,10,17)(11,19,18,16,15,20)(12,14,13);(2,4)(6,8)(9,11,12,16)(10,18,19,17,14,20,15,13)

Polycyclic presentation:

2^3,3,2^6|b^2=c,e^2=i,f^2=j,b^a=bc,d^a=d^2,d^b=dj,e^a=fi,e^b=ei,e^d=fi,f^a=ej,f^b=fj,f^d=ef,g^a=ghi,g^d=h,h^a=hij,h^d=gh,i^a=j,i^d=j,j^a=i,j^d=ij

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267836
maximal normal subgroup of#72267836
maximal quotient of#72267840

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);